Wave Mechanics: Standing Waves

WM09 introduced the principle of superposition. A particularly important application occurs when two sinusoidal waves of equal amplitude, equal frequency, and equal wavelength travel through the same region in opposite directions. Their sum does not look like a sinusoid translating steadily to the right or left. Instead, the pattern contains fixed positions that never move and other positions that oscillate with maximum amplitude.

This pattern is called a standing wave. Standing waves arise in many physical systems, including stretched strings, air columns, mechanical structures, electromagnetic cavities, and quantum wave problems. Standard introductory treatments derive them from the superposition of oppositely traveling waves [1, 2, 3].

The central goal of WM10 is to derive the standing-wave form and understand its geometry before later lessons impose boundary conditions and select allowed modes.

1 Start with two equal waves traveling in opposite directions

Consider the right-moving wave

u (x, t) = A cos(kx − ωt) 1

(1)

and the left-moving wave

u2(x,t) = A cos(kx + ωt ).

(2)

They have the same amplitude A, Wavenumber k, and angular frequency ω. Their only essential difference is their direction of propagation.

By superposition,

u (x,t) = u1 (x,t) + u2(x, t).

(3)

Using the identity

cos(α − β ) + cos(α + β) = 2cos αcos β,

(4)

with

α = kx, β = ωt,

(5)

we obtain

|----------------------------| |u(x,t) = 2A cos(kx) cos(ωt).| -----------------------------

(6)

This is a standing wave.

Figure. Two equal sinusoidal waves traveling in opposite directions superpose to form a standing-wave profile. The component waves translate, but the locations of the nodes and antinodes of their sum remain fixed.

OpenStax gives the same physical construction: two identical waves moving in opposite directions alternately interfere constructively and destructively, producing a resultant pattern that does not propagate through space [3].

2 Why the result is not a traveling wave

A traveling wave such as

A cos(kx − ωt)

(7)

contains space and time inside one phase combination. The whole pattern shifts with time.

The standing-wave expression is different:

u(x,t) = 2A cos(kx) cos(ωt ) . ◟---◝◜---◞ ◟-◝◜--◞ spatial factor temporal factor

(8)

The dependence on x and t is separated into a product. The spatial factor determines the oscillation amplitude available at each location. The temporal factor causes the pattern to oscillate in place.

The spatial envelope does not translate. Instead, the entire pattern passes through a sequence such as

+maximum shape − → 0 − → − maximum shape − → 0 −→ +maximum shape.

(9)

Figure. Snapshots of one standing wave at four times during a cycle. The pattern does not translate horizontally. The same node positions remain fixed while the lobes reverse sign through the cycle.

A standing wave therefore does not mean that the medium is motionless. Except at special fixed points called nodes, the medium can oscillate strongly. What stands still is the spatial interference pattern.

3 Local amplitude

At a fixed position x = x₀,

u(x ,t) = 2A cos(kx ) cos(ωt). 0 0

(10)

This is simple harmonic motion in time. The magnitude of its local amplitude is

|------------------------| -Alocal(x)-=-2A-|cos(kx)|.|

(11)

The absolute value is important. A negative value of cos(kx) does not mean that an amplitude is physically negative. It means that the temporal motion at that position is shifted by π relative to a neighboring region where cos(kx) is positive.

Thus a standing wave is an entire continuum of oscillators, all with the same angular frequency ω, but with position-dependent amplitudes and with neighboring lobes alternating in temporal phase.

4 Nodes

A node is a position that remains at zero displacement for all time. For

u(x,t) = 2A cos(kx) cos(ωt),

(12)

this requires

cos(kx ) = 0.

(13)

Therefore

π- kx = 2 + nπ, n = 0, 1,2,...

(14)

and

x_node	=	(15)
	= .	(16)

Thus, for this particular choice of spatial phase,

|------------------| |x = (2n-+--1)λ.| --node--------4------

(17)

Every node remains fixed because its spatial factor is permanently zero.

5 Antinodes

An antinode is a position at which the magnitude of the standing-wave amplitude is maximum. We require

|cos(kx)| = 1.

(18)

Hence

kx = nπ

(19)

and

|--------------| | n-λ | |xantinode = 2 .| ----------------

(20)

At an antinode, the maximum displacement magnitude is

|----| -2A.-|

(21)

This is twice the amplitude of either component traveling wave because the two components interfere constructively there at the times of maximum standing-wave displacement.

6 Node and antinode spacing

The exact coordinate of a node depends on the spatial phase convention used to write the standing wave. The spacings, however, are invariant.

Adjacent nodes are separated by

|--| |λ | |-.| -2--

(22)

Adjacent antinodes are also separated by

|--| |λ-| |2.| ----

(23)

A node and its nearest antinode are separated by

|--| |λ-| |4.| ----

(24)

Figure. Node and antinode geometry for 2A cos(kx) cos(ωt). Adjacent nodes are one-half wavelength apart, adjacent antinodes are one-half wavelength apart, and a node is one-quarter wavelength from its nearest antinode.

These spacing rules are often the fastest way to infer wavelength from an observed standing-wave pattern.

7 Equivalent standing-wave forms

The form

2A cos(kx) cos (ωt )

(25)

is not the only possible representation. Different choices of spatial and temporal phase can produce forms such as

2A sin(kx)cos(ωt )

(26)

2A sin (kx )sin(ωt).

(27)

These equations describe the same general standing-wave structure with the origin or time reference shifted.

For example,

u (x, t) = 2A sin(kx )cos(ωt)

(28)

has a node at x = 0 because sin 0 = 0. This form is especially convenient when describing a string fixed at the origin. Feynman’s discussion of confined waves uses exactly this idea: boundary conditions force nodes at fixed locations and thereby restrict the permitted values of k [4].

WM10 does not yet develop the full boundary-value problem. The important point for now is that the choice between sine and cosine changes the coordinates of the nodes but not the physical spacing between them.

8 Adjacent loops move in opposite temporal phase

Consider two positions in neighboring lobes of the standing wave. Their spatial factors cos(kx) have opposite signs. Therefore, when one lobe has positive displacement, the neighboring lobe has negative displacement.

Their time dependence differs effectively by a phase shift of π.

Points within the same lobe move in the same temporal phase, although with different amplitudes. Crossing a node changes the sign of the spatial factor, so the next lobe moves oppositely.

Figure. Two snapshots separated by one-half period. Each lobe reverses sign, while neighboring lobes remain opposite in temporal phase. The nodes stay fixed at zero displacement throughout the motion.

9 How standing waves are produced physically

The algebra began with two equal counter-propagating waves. A common physical way to obtain such a pair is reflection. A traveling wave reaches a boundary, is reflected, and overlaps the incoming wave. If the incident and reflected waves have the appropriate amplitudes, frequencies, wavelengths, and phases, their superposition can produce fixed nodes and antinodes.

In finite systems, the boundaries usually impose additional restrictions. A string fixed at both ends, for example, must have a node at each end. Only certain wavelengths can satisfy both conditions simultaneously. Those special patterns are normal modes. OpenStax and MIT’s 8.03 course develop this connection between standing waves, boundaries, normal modes, and resonance [3, 5].

Those boundary-selected modes are a major subject in their own right and are reserved for later articles. WM10 focuses on the standing-wave structure that exists before those restrictions are imposed.

10 Perfect standing waves require matching counter-propagating components

Fixed nodes arise cleanly when the two component waves have the same frequency, wavelength, and amplitude and propagate in opposite directions.

If the frequencies differ, the interference pattern changes with time rather than remaining stationary. If the amplitudes differ, complete cancellation at would-be nodes generally does not occur. Thus the ideal standing-wave form is a special, highly organized superposition rather than an arbitrary overlap of two waves.

11 Standing waves and energy transport

A traveling wave has a clear direction of propagation. A standing-wave pattern does not translate in one direction. This does not mean that energy or motion is absent. The medium can oscillate substantially at antinodes, and energy can exchange locally between different forms.

A careful treatment of energy density, power, and net energy flux requires additional machinery and is deferred to the later energy-and-power portion of the Wave mechanics series. For WM10, the essential statement is only that the standing-wave pattern itself has no net direction of translation.

12 Worked example 1: locate nodes and antinodes

A standing wave is

u(x,t) = 6.0mm cos(4 πx)cos(20πt ),

(29)

where x is measured in meters and t in seconds.

The wavenumber is

k = 4π rad/m.

(30)

Therefore

λ	=	(31)
	= m	(32)
	= 0.50 m .	(33)

Adjacent antinodes are separated by

λ |------| 2- = -0.25-m-,

(34)

and a node is one-quarter wavelength from its nearest antinode:

λ |--------| --= -0.125m--. 4

(35)

Since the cosine standing wave has an antinode at x = 0, the first positive node occurs at

|------------| x-=--0.125m--.

(36)

13 Worked example 2: recover the component waves

Suppose

u(x,t) = 10 mm cos(kx) cos(ωt ).

(37)

Comparing this with

u = 2A cos(kx)cos(ωt),

(38)

we identify

2A = 10 mm.

(39)

Thus each traveling component has amplitude

|------------| A = 5.0mm . -------------

(40)

One possible pair of component waves is therefore

u₁	= 5.0 mm cos(kx − ωt),	(41)
u₂	= 5.0 mm cos(kx + ωt).	(42)

Their superposition reproduces the standing wave exactly.

14 Worked example 3: infer wavelength from node spacing

A laboratory standing-wave pattern has adjacent nodes separated by

0.18 m.

(43)

Because adjacent nodes are separated by λ∕2,

	= 0.18 m,	(44)
λ	= 0.36 m .	(45)

The nearest antinode to either node lies halfway between them, so the node-to-antinode spacing is

λ |--------| --= -0.090m--. 4

(46)

15 Common mistakes

Mistake: thinking a standing wave means nothing moves. Nodes do not move, but points between nodes generally oscillate.
Mistake: measuring a wavelength from one node to the next. Adjacent nodes are separated by λ∕2, not λ.
Mistake: calling 2A cos(kx) the amplitude without qualification. Its sign changes. The physical local amplitude magnitude is 2A| cos(kx)|.
Mistake: assuming every pair of opposite-going waves makes a perfect standing wave. The ideal stationary-node pattern requires matched frequency, wavelength, and amplitude.
Mistake: assuming the sine and cosine standing-wave forms describe different physics. They may simply correspond to different choices of spatial or temporal origin.
Mistake: assuming a standing wave is already a normal mode. Boundary conditions must still determine which standing-wave patterns are permitted in a finite system.

16 What WM10 adds to the wave-mechanics language

WM09 established that linear waves add. WM10 applies that principle to two matched waves traveling in opposite directions:

A cos(kx − ωt) + A cos(kx + ωt)

= 2A cos(kx) cos(ωt) .

(47)

The resulting standing wave has fixed nodes and antinodes. For the cosine form,

(2n-+-1)λ- nλ- xnode = 4 , xantinode = 2 .

(48)

Regardless of the phase convention,

|------------------| |node-to-node = λ-, -----------------2-|

(49)

|-------------------------| | λ | |antinode-to-antinode = --|, ------------------------2--

(50)

and

|-----------------------------| | λ-| |node-to-nearest-antinode = 4 . -------------------------------

(51)

These ideas prepare the way for boundary conditions, resonance, normal modes, and eventually Fourier and eigenfunction methods.

17 References

References

[1] A. P. French, Vibrations and Waves, M.I.T. Introductory Physics Series, W. W. Norton & Company, 1971.

[2] Frank S. Crawford, Jr., Waves, Berkeley Physics Course, Volume 3, McGraw-Hill, 1968.

[3] William Moebs, Samuel J. Ling, and Jeff Sanny, University Physics, Volume 1, OpenStax, 2016, Section 16.6, “Standing Waves and Resonance.”

[4] Richard P. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics, Volume I, Chapter 49, “Modes,” especially the discussion of reflected waves, nodes, and confined standing-wave patterns.

[5] Massachusetts Institute of Technology, 8.03SC Physics III: Vibrations and Waves, Lecture 9, “Wave Equation, Standing Waves, Fourier Series,” Fall 2016, MIT OpenCourseWare.

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